Question

A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 9 feet below the level of the pulley.

If the rope is pulled through the pulley at a rate of 10 ft/min, at what rate will the boat be approaching the dock when 90 ft of rope is out?

Answer 1

The horizontal and vertical distances from the pulley form a right angle triangle, subtending an angle of ∅.
We calculate ∅ using:
tan(∅) = y/x
∅ = tan⁻¹(9/90)
∅ = 5.71°
The horizontal component of the boat's velocity can be obtained using:
Vx = Vcos(∅)
Vx = 10cos(5.71)
Vx = 9.95 ft/min

The boat approaches the dock at a rate of 9.95 ft/min